Exact Solution of Noncommutative U(1) Gauge Theory in $4-$Dimensions

Noncommutative U(1) gauge theory on the Moyal-Weyl space ${\bf R}^2{\times}{\bf R}^2_{\theta}$ is regularized by approximating the noncommutative spatial slice ${\bf R}^2_{\theta}$ by a fuzzy sphere of matrix size $L$ and radius $R$ . Classically we observe that the field theory on the fuzzy space ${\bf R}^2{\times}{\bf S}^2_L$ reduces to the field theory on the Moyal-Weyl plane ${\bf R}^2{\times}{\bf R}^2_{\theta}$ in the flattening continuum planar limits $R,L{\longrightarrow}{\infty}$ where $R^2/(|L|^2)^q={\theta}^2$ and $q>{3/2}$ . The effective noncommutativity parameter is found to be given by ${\theta}_{eff}^2{\sim}2{\theta}^2(\frac{L}{2})^{2q-1}$ and thus it corresponds to a strongly noncommuting space. In the quantum theory it turns out that this prescription is also equivalent to a dimensional reduction of the model where the noncommutative U(1) gauge theory in 4 dimensions is shown to be equivalent in the large $L$ limit to an ordinary $O(M)$ non-linear sigma model in 2 dimensions where $M{\sim}3L^2$ . The Moyal-Weyl model defined this way is also seen to be an ordinary renormalizable theory which can be solved exactly using the method of steepest descents . More precisely we find for a fixed renormalization scale $\mu$ and a fixed renormalized coupling constant $g_r^2$ an $O(M)-$symmetric mass, for the different components of the sigma field, which is non-zero for all values of $g_r^2$ and hence the $O(M)$ symmetry is never broken in this solution . We obtain also an exact representation of the beta function of the theory which agrees with the known one-loop perturbative result